# The following table shows the marks scored by 140 students in an examination of a certain paper:  Marks: 0 - 10, 10 - 20, 20 - 30, 30 - 40, 40 - 50

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The following table shows the marks scored by 140 students in an examination of a certain paper:

 Marks: 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 Number of students: 20 24 40 36 20

Calculate the average marks by using all the three methods: direct method, assumed mean deviation and shortcut method.

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From Direct method:

 Class interval Mid value (xi) fi fixi 0 - 10 5 20 100 10 - 20 15 24 360 20 - 30 25 40 1000 40 - 50 45 20 900 N = 140 $\sum$fixi = 3620

Mean = $\frac{\sum f_ix_i}{N}$

$=\frac{3620}{140}=25.857$

Assumed mean method:

let assumed mean (A) = 25

Mean = A + $\frac{\sum f_iu_i}{N}$

 Class interval Mid value (xi) ui = xi - A fi fiui 0 - 10 5 -20 20 -400 10 - 20 15 -10 24 -240 20 - 30 25 0 40 0 30 - 40 35 10 36 360 40 - 50 45 20 20 400 N = 140 $\sum$fiui = 120

Mean = A + $\frac{\sum f_ix_i}{N}$

$=25+\frac{120}{140}=25+0.857$

$=25.857$

Step deviation method: Let the assumed mean (A) = 25

 Class interval Mid value (xi) di = xi - A = xi - 25 ui = $\frac{xi-25}{10}$ Frequency (fi) fiui 0 - 10 5 -20 -2 20 -40 10 - 20 15 -10 -1 24 -24 20 - 30 25 0 0 40 0 30 - 40 35 10 1 36 36 40 - 50 45 20 2 20 40 N = 140 $\sum$fiui = 12

Mean = A + $\frac{\sum f_iu_i}{N}\times h$

$=25+0.857=25.857$

$=25+\frac{12}{140}\times10$

$=25+0.857=25.857$